This is a style of question that people use mostly in interviews. Asking a question like this will make the listener answer more slowly and carefully:

What would you say was the best part about working there?

What would you say is your favorite film of the last 10 years?

There can also be questions of this type that start with different question words:

Who would you say is the strongest member of your team?

You might also see questions like these on written surveys.